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A MIXED FINITE ELEMENT METHOD FOR A SIXTH ORDER ELLIPTIC PROBLEM

JÉRÔME DRONIOU∗, MUHAMMAD ILYAS†† , BISHNU P. LAMICHHANE† , AND GLEN E. WHEELER‡

Abstract. We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the H1-conforming Lagrange finite element spaces to approximate the solution. We prove a priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods.

Key words. Sixth order problem, higher order partial differential equations, biharmonic problem, mixed finite elements, error estimates.

AMS subject classifications. 65N30, 65N15, 35J35 (Primary) 35J40 (Secondary)

1. Introduction. Partial differential equations (PDE) have a long and rich history of application in physical problems. One of their main advantages is in the modelling of ideal or desired structures [36]. In particular, one may wish to fill a curve with a solid material that satisfies certain conditions along the boundary. Depending on the application, there may be several constraints along the curve. In many applications these filled curves (called components) are fitted together to form a larger shape. It is natural and in some situations essential that at least some of the derivatives of the surface are continuous across the boundary curves.

In this context, higher-order partial differential equations come to the fore: for a solution of a partial differential equation of order 2k, one may typically allow restrictions on all derivatives up to order (k − 1) along the boundary curve. This guarantees their continuity across components.

Continuity of the second derivative across boundaries, achieved by the sixth-order PDE proposed in this article, is critical in several settings. In the construction of automobiles, each panel is designed by a computer based on given specifications. Aesthetics are an important aspect, and in this regard, the composition of reflections from the surface of a car panel must be considered. If one prescribes only the derivatives up to first order along the boundary, then this leaves open the possibility of the second derivative of the panel changing sign across the boundary. In practical terms, this causes boundaries to move from being convex to concave, or vice-versa. Reflections will flip across such boundaries, which from an aesthetic perspective is unacceptable.

The strength and maximal load bearing of tensile structures also depends critically on the conti- nuity of higher derivatives across component boundaries. Force is optimally spread uniformly across components, however, where derivatives of the surface are large, force and load are accumulated. This can be by design. It is dangerous however when force accumulates across a boundary due not to de- sign but to a discontinuity in one of the higher derivatives across that boundary. This concern can be alleviated when a number of derivatives dependent upon the total expected load of the structure can be guaranteed to be continuous. Two derivatives are guaranteed by our scheme and this is typically enough for most minor structures, such as small buildings, residential homes, and vehicles.

Sixth-order PDE have arisen in a variety of other contexts, from propeller blade design [13] to ulcer modelling [33]. Generic applications of sixth-order PDE to manufacturing are mentioned in

∗School of Mathematical Sciences, Monash University, Victoria 3800, Australia. email: jerome.droniou@monash.edu †School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308,

Bishnu.Lamichhane@newcastle.edu.au, ‡Institute for Mathematics and its Applications, School Statistics, University of Wollongong, glenw@uow.edu.au

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[3, 5]. Applications of sixth order problems in surface modelling and fluid flows are considered in [25, 31].

To see that sixth-order PDE are natural for such applications, it is instructive to view such an equation variationally. Minimising the classical Dirichlet energy, we calculate the first variation of the functional ∫

Ω

|∇u|2dx ,

and find the Laplace equation

∆u = 0

or, in the case of the gradient flow, the heat equation

(∂t −∆)u = 0 .

Minimising the elastic energy, the integrand of the functional to be minimised depends on an additional order of derivative of u, and so the Euler-Lagrange equation and resulting gradient flow is of fourth- order. If we are additionally interested in minimising the rate of change of curvature across the surface, the ‘rate of change of acceleration’ or jerk, then the functional will depend on three orders of derivatives of u. The resulting Euler-Lagrange equation

∆3u = 0

and gradient flow

(∂t −∆3)u = 0

depend on six orders of derivatives of u. This perspective is taken in Section 2, where the variational formulation is made rigorous. Recent resarch interest in such equations includes [18, 19, 26, 28].

In geophysics, sixth-order PDE are used to overcome difficulties involving complex geological faults [35]. Indeed, sixth-order PDE arise in a variety of geophysical contexts due to their appearance as models in electromagneto-thermoelasticity [30] and relation to equatorial electrojets [34]. We remark that model PDE from geophysics are in general quite interesting to study from a PDE perspective, with issues such as non-uniqueness and general ill-posedness fundamental characteristics; we refer to [24] for a selection of such issues.

The major contribution in our paper is a mixed finite element scheme for a sixth-order partial differential equation. This allows one to accurately model components arising from prescribed (up to and including) second order derivatives along boundary curves. Another approach to approximate the solution of the sixth-order elliptic problem based on the interior penalty is considered by Gudi and Neilan [17]. In Section 2 we introduce our setting, which considers two different sets of boundary conditions: simply supported, and clamped. We use constrained minimisation to cast our problems in a mixed formulation as in the case of the biharmonic equation [9, 12, 21] (other approaches to mixed formulations for the biharmonic equation can be found in [11, 10, 14, 15, 2, 27, 22]). The resulting saddle point problem allows us to apply low order H1-conforming finite element methods to approximate the solution of the sixth order problem. This approximation is described, for both sets of boundary conditions, in Section 3. A-priori error estimates are proved in Section 4. The optimality of the predicted rates of convergences is illustrated, for each boundary condition, in Section 5 through various numerical results.

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2. A mixed formulation of a sixth order elliptic equation. Let Ω ⊂ Rd, d ∈ {2, 3}, be a bounded domain with polygonal or polyhedral boundary ∂Ω and outward pointing normal n on ∂Ω. We consider the sixth order problem

−∆3u = f in Ω (2.1)

with f ∈ H−1(Ω) and two sets of boundary conditions (BCs). The first set is the set of simply supported boundary conditions

u = ∆u = ∆2u = 0 on ∂Ω, (2.2)

and the second set is the set of clamped boundary conditions

u = ∂u

∂n = ∆u = 0 on ∂Ω. (2.3)

We aim at obtaining a formulation only based on the H1-Sobolev space. We begin by defining the Lagrange multiplier space:

• Simply supported boundary conditions. We set

Mbc = H 1 0 (Ω) ,

and equip Mbc with the norm

‖v‖Mbc = ‖v‖1,Ω .

• Clamped boundary conditions. We set

Mbc = {q ∈ H−1(Ω) : ∆q ∈ H−1(Ω)} ,

where ∆q is interpreted in the distributional sense, and the space Mbc is equipped with the graph norm

‖q‖Mbc = √ ‖q‖2−1,Ω + ‖∆q‖2−1,Ω.

We use the notation 〈·, ·〉 for the duality pairing between the two spaces H10 (Ω) and H−1(Ω), so that 〈u, q〉 and 〈u,∆q〉 are well defined for u ∈ H10 (Ω) and q ∈ Mbc. We note that this space Mbc is less regular than H

1(Ω), compare, e.g., [4, 37].

Let k ∈ N∪{0}. We use the standard notations to represent Sobolev spaces [1, 8]. We use (·, ·)k,Ω and ‖ · ‖k,Ω to denote the inner product and norm in Hk(Ω), respectively. When k = 0, we get the inner product (·, ·)0,Ω and the norm ‖ · ‖0,Ω in L2(Ω). The norm of W k,p(Ω) is denoted by ‖ · ‖k,p,Ω.

To obtain the H1-based formulation of our boundary value problems, we introduce an additional unknown φ = ∆u and write a weak form of this equation by formally multiplying by a function q ∈Mbc and integrating over Ω, as in [4, 37]. The variational equation is now written as

〈φ, q〉 − 〈u,∆q〉 = 0, q ∈Mbc.

Keeping in mind that u will be taken in H10 (Ω), and considering the BC-dependent Mbc, we see that this variational definition of “φ = ∆u” also formally imposes the condition ∂u∂n = 0 on ∂Ω, in the case of clamped BCs. For simply supported BCs, this does not impose any additional boundary conditions.

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To write the mixed formulation in a standard setting, we introduce the function space V = H10 (Ω)×H10 (Ω) with the inner product (·, ·)V defined as

((u, φ), (v, ψ))V = (∇u,∇v)0,Ω + (∇φ,∇ψ)0,Ω

and with the norm ‖ · ‖V induced by this